Complex Numbers

Basics of Complex Numbers and Algebra

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Basics of Complex Numbers and Algebra

Basics of Complex Numbers and Algebra

This lesson covers the following key concepts:

  • Definition of complex numbers: z = a + ib where i² = -1
  • Real and imaginary parts of complex numbers
  • Equality of complex numbers
  • Algebra: addition, subtraction, multiplication, division
  • Complex conjugate and its properties
  • Multiplicative inverse of complex numbers

Important Formulas

  • z=a+ib where i2=−1z = a + ib \text{ where } i^2 = -1
  • z‾=a−ib (conjugate)\overline{z} = a - ib \text{ (conjugate)}
  • z+z‾=2Re(z),z−z‾=2iIm(z)z + \overline{z} = 2\text{Re}(z), z - \overline{z} = 2i\text{Im}(z)
  • z⋅z‾=a2+b2=∣z∣2z \cdot \overline{z} = a^2 + b^2 = |z|^2
  • 1z=z‾∣z∣2\frac{1}{z} = \frac{\overline{z}}{|z|^2}
  • z1+z2‾=z1‾+z2‾,z1z2‾=z1‾⋅z2‾\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}, \overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}